1. Addition of Angular Momentum
We’ve learned that angular momentum is important in quantum mechanics
Orbital angular momentum L
Spin angular momentum S
For multielectron atoms, we need to learn to add angular momentum
Multiple electrons, each with li and si
Spin-orbit interaction couples L and S to form a total angular momentum J

2. Addition of Angular Momentum
What is the total spin of two electrons?
Again we must take into account that electrons are identical particles

3. Addition of Angular Momentum
Comments

4. Addition of Angular Momentum
Comments
The symmetry properties under particle interchange of the wave function for fermions and bosons refers to the TOTAL wave function ψ(space)χ(spin)
For fermions, since the overall wave function must be antisymmetric under interchange of particles 1 and 2, we can have either

5. Addition of Angular Momenta
The complete wave functions are
The symmetric spin function could be any of the S=1 triplet states
Note that for the S=1 triplet state the spatial wave function has low probability for x1~ x2
Parallel spins repel
Note that for the S=0 singlet state the spatial wave function has high probability for x1~ x2
Opposite spins attract
This spin pairing arises from the exchange “force” we talked about earlier

6. Addition of Angular Momentum
Comments
Look at the He atom again (1s2)
What is the spin of the ground state of He?
What is the spin of the first excited state of He?
Estimate the ground state energy of He
Estimate for the first excited state of He

7. Addition of Angular Momentum
Comments
Based on this example, the rules for addition of (spin) angular momentum are
The same rules hold for the addition of any two (or more) angular momenta

8. Spin-Orbit Interaction
Magnetic field produced by orbiting proton (nuclear dipole moment)

9. Spin Orbit Interaction
Quick review of orbital and spin magnetic moments

10. Spin-Orbit Interaction
This internal magnetic field gives rise to a spin-orbit interaction term in the Hamiltonian

11. Spin-Orbit Interaction
Actually this isn’t quite right since the electron is in a non-inertial frame
The correct result differs by a factor of ½ and is know as the Thomas precession

12. Spin-Orbit Interaction
Aside

13. Spin-Orbit Interaction
Aside,

14. Spin-Orbit Interaction
We left the spin-orbit interaction out of the Hamiltonian the first time we did the hydrogen atom
In part, because it’s a small correction
The full Schrodinger equation for the hydrogen atom is (neglecting a relativistic correction)

15. Spin-Orbit Interaction
Our original quantum numbers for the hydrogen atom were
And
But with the spin-orbit term in the Hamiltonian, H no longer commutes with some of these operators
Before you answer note that

16. Total Angular Momentum
The spin-orbit interaction couples the orbital (L) and spin (S) angular momentum to form the total angular momentum (J)
The internal magnetic field is determined by L and this acts on the spin magnetic dipole of the electron determined by S so the two angular momenta are not independent
The new “good” quantum numbers of the hydrogen atom are n, l, s, j, mj

17. Total Angular Momentum
Coupling of L and S to form J
L and S precess around J
J precesses around the z axis